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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Probability.ContinuumTransition.KacEnergy
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
theorem familyTerm_product_bound (F G H : ConfigFamily)
(hG : RelabelInvariant G) (hH : RelabelInvariant H) {A B : Set Space}
(hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B)
(hGF : GrandFinite G A) (hHF : GrandFinite H B) (n : ℕ)
(hFi : IntegrableOn (F (Fin n)) {x | ∀ i, x i ∈ A ∪ B})
(hbound : ∀ (s : Finset (Fin n)) (x : Fin n → Space),
x ∈ binaryRegion A B s → F (Fin n) x ≤
G {i : Fin n // i ∈ s} (fun i => x i)*H {i : Fin n // i ∉ s} (fun i => x i)) :
familyTerm F (A ∪ B) n ≤ ∑ k ∈ Finset.range (n+1), familyTerm G A k*familyTerm H B (n-k) := by
have hb := integral_binary_bound hA hB hd
(fun k => familyIntegral G A (Fin k)) (fun k => familyIntegral H B (Fin k))
hFi ?_
swap
· intro s
have hprod := integrableOn_binary_product volume A B s
(by simpa only [← volume_pi] using hGF.integrable {i : Fin n // i ∈ s})
(by simpa only [← volume_pi] using hHF.integrable {i : Fin n // i ∉ s})
have hi := setIntegral_mono_on
(hFi.mono_set (binaryRegion_subset_union A B s))
(by simpa only [← volume_pi] using hprod)
(measurableSet_binaryRegion hA hB s) (hbound s)
rw [show volume = Measure.pi (fun _ : Fin n => (volume : Measure Space)) by exact volume_pi,
setIntegral_binary_product volume A B s (G {i : Fin n // i ∈ s})
(H {i : Fin n // i ∉ s})] at hi
simp only [← volume_pi] at hi
have heg := familyIntegral_eq_Fin G hG A {i : Fin n // i ∈ s}
have heh := familyIntegral_eq_Fin H hH B {i : Fin n // i ∉ s}
have hc : Fintype.card {i : Fin n // i ∈ s} = s.card := Fintype.card_coe s
rw [hc] at heg
have hcc : Fintype.card {i : Fin n // i ∉ s} = n-s.card := by
have hh := Fintype.card_subtype_compl (fun i : Fin n => i ∈ s)
simpa only [Fintype.card_fin, hc] using hh
rw [hcc] at heh
change (∫ x in binaryRegion A B s, F (Fin n) x) ≤ _
calc
_ ≤ familyIntegral G A {i : Fin n // i ∈ s}*
familyIntegral H B {i : Fin n // i ∉ s} := hi
_ = _ := by rw [heg, heh, Fintype.card_fin]
have hh := div_le_div_of_nonneg_right hb (Nat.cast_nonneg (n.factorial) : (0 : ℝ) ≤ n.factorial)
rw [Fintype.card_fin, factorial_binomial_sum
(fun k => familyIntegral G A (Fin k)) (fun k => familyIntegral H B (Fin k)) n] at hh
exact hh
theorem GrandFinite.union_product {F G H : ConfigFamily}
(hG : RelabelInvariant G) (hH : RelabelInvariant H) {A B : Set Space}
(hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B)
(hGF : GrandFinite G A) (hHF : GrandFinite H B)
(hFi : ∀ (ι : Type) [Fintype ι], IntegrableOn (F ι) {x | ∀ i, x i ∈ A ∪ B})
(h0 : ∀ (ι : Type) [Fintype ι] (x : ι → Space), 0 ≤ F ι x)
(hbound : ∀ (n : ℕ) (s : Finset (Fin n)) (x : Fin n → Space),
x ∈ binaryRegion A B s → F (Fin n) x ≤
G {i : Fin n // i ∈ s} (fun i => x i)*H {i : Fin n // i ∉ s} (fun i => x i)) :
GrandFinite F (A ∪ B) := by
refine ⟨hFi, ?_⟩
have hab : Summable (fun p : ℕ × ℕ => familyTerm G A p.1*familyTerm H B p.2) :=
summable_mul_of_summable_norm (summable_norm_iff.mpr hGF.summable)
(summable_norm_iff.mpr hHF.summable)
exact Summable.of_nonneg_of_le (familyTerm_nonneg F (A ∪ B) h0)
(fun n => familyTerm_product_bound F G H hG hH hA hB hd hGF hHF n (hFi (Fin n)) (hbound n))
(summable_sum_mul_range_of_summable_mul hab)
end ContinuumTemperature
end
end OAI
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